3.14.3 \(\int \frac {3+5 x}{1-2 x} \, dx\)

Optimal. Leaf size=16 \[ -\frac {5 x}{2}-\frac {11}{4} \log (1-2 x) \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \begin {gather*} -\frac {5 x}{2}-\frac {11}{4} \log (1-2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)/(1 - 2*x),x]

[Out]

(-5*x)/2 - (11*Log[1 - 2*x])/4

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {3+5 x}{1-2 x} \, dx &=\int \left (-\frac {5}{2}-\frac {11}{2 (-1+2 x)}\right ) \, dx\\ &=-\frac {5 x}{2}-\frac {11}{4} \log (1-2 x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.06 \begin {gather*} \frac {1}{4} (-10 x-11 \log (1-2 x)+5) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)/(1 - 2*x),x]

[Out]

(5 - 10*x - 11*Log[1 - 2*x])/4

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3+5 x}{1-2 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + 5*x)/(1 - 2*x),x]

[Out]

IntegrateAlgebraic[(3 + 5*x)/(1 - 2*x), x]

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fricas [A]  time = 0.99, size = 12, normalized size = 0.75 \begin {gather*} -\frac {5}{2} \, x - \frac {11}{4} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x),x, algorithm="fricas")

[Out]

-5/2*x - 11/4*log(2*x - 1)

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giac [A]  time = 1.12, size = 13, normalized size = 0.81 \begin {gather*} -\frac {5}{2} \, x - \frac {11}{4} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x),x, algorithm="giac")

[Out]

-5/2*x - 11/4*log(abs(2*x - 1))

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maple [A]  time = 0.00, size = 13, normalized size = 0.81 \begin {gather*} -\frac {5 x}{2}-\frac {11 \ln \left (2 x -1\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)/(1-2*x),x)

[Out]

-5/2*x-11/4*ln(2*x-1)

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maxima [A]  time = 0.52, size = 12, normalized size = 0.75 \begin {gather*} -\frac {5}{2} \, x - \frac {11}{4} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x),x, algorithm="maxima")

[Out]

-5/2*x - 11/4*log(2*x - 1)

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mupad [B]  time = 0.03, size = 10, normalized size = 0.62 \begin {gather*} -\frac {5\,x}{2}-\frac {11\,\ln \left (x-\frac {1}{2}\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x + 3)/(2*x - 1),x)

[Out]

- (5*x)/2 - (11*log(x - 1/2))/4

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sympy [A]  time = 0.08, size = 15, normalized size = 0.94 \begin {gather*} - \frac {5 x}{2} - \frac {11 \log {\left (2 x - 1 \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x),x)

[Out]

-5*x/2 - 11*log(2*x - 1)/4

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